Holomorphic symplectic and Poisson structures on the holomorphic cotangent bundle of a complex Lie group and of a holomorphic principal bundle (Q2868576)

The author studies the holomorphic symplectic and Poisson structures defined on the holomorphic cotangent bundle of a complex Lie group and of a holomorphic principal bundle obtaining some results similar to those from \textit{D. Alekseevsky} et al. [J. Math. Phys. 35, No. 9, 4909--4927 (1994; Zbl 0822.58016)]. The author shows how the techniques of obtaining symplectic structures and Poisson structures on the cotangent bundle of a Lie group or a principal bundle can be applied in the holomorphic cases. After presenting some basic facts about holomorphic fibrations, holomorphic semi-basic \(1\)-forms, holomorphic symplectic \(2\)-forms, and holomorphic Poisson structures, the author defines a holomorphic Liouville \(1\)-form on a holomorphic symplectic manifold \((E,\omega)\), fibered over a complex manifold \(M\) with holomorphic Lagrangian fibers. This is a horizontal \(1\)-form \(\theta\) with \(\omega =\partial \theta\). The author considers the Liouville lift which assigns to a holomorphic vector field \(Z\in \chi (M)\) the holomorphic Hamiltonian vector field \(Z^\theta\) on \(E\) for the holomorphic function \(-\theta (Z\circ \pi)\), where \(\pi:(E,\omega)\to M\) is the holomorphic projection. Using the Liouville lift, one obtains the explicit expression of the holomorphic Poisson structure \(\Lambda =\omega^{- 1}\in \Gamma (\bigwedge^2T^{1,0}E)\). Next, by applying the above construction to the holomorphic cotangent bundle \(\pi:(T^{1,0}G)^*\to G\) of a complex Lie group \(G\), the author obtains the standard holomorphic symplectic form \(\omega\) on \((T^{1,0}G)^*\). Then he obtains the associated holomorphic Poisson structure \(\Lambda\) on \((T^{1,0}G)^*\). A similar construction is made in the case of a holomorphic principal \(G\)-bundle, \(p:P\to M\).